Abstract A partial solution of the quaternionic contact Yamabe problem on the quater- nionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Bi- quard connection is zero and this occurs precisely on 3-Sasakian manifolds. All con- formal transformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact struc- ture with vanishing torsion of the Biquard connection are explicitly described. A ‘3-Hamiltonian form’ of infinitesimal conformal automorphisms of quaternionic con- tact structures is presented. Received by the editor September 11, 2007. Article electronically published on January 31, 2014. DOI: http://dx.doi.org/10.1090/memo/1086 2010 Mathematics Subject Classification. Primary 53C17. Key words and phrases. Yamabe equation, quaternionic contact structures, Einstein structures. This project has been funded in part by the National Academy of Sciences under the [Collab- oration in Basic Science and Engineering Program 1 Twinning Program] supported by Contract No. INT-0002341 from the National Science Foundation. The contents of this publication do not necessarily reflect the views or policies of the National Academy of Sciences or the National Sci- ence Foundation, nor does mention of trade names, commercial products or organizations imply endorsement by the National Academy of Sciences or the National Science Foundation. Affiliations at time of publication: Stefan Ivanov, University of Sofia and Institute of Math- ematics, Bulgarian Academy of Sciences, Faculty of Mathematics and Informatics, blvd. James Bourchier 5, 1164, Sofia, Bulgaria, email: ivanovsp@fmi.uni-sofia.bg Ivan Minchev, University of Sofia, Faculty of Mathematics and Informatics, blvd. James Bourchier 5, 1164 Sofia, Bul- garia Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 61137 Brno, Czech Republic, email: minchev@fmi.uni-sofia.bg and Dimiter Vassilev, Department of Mathe- matics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131-0001, email: vassilev@math.unm.edu. c 2014 American Mathematical Society v
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